Title: Polynomial functors and trees

 Abstract: The aim of this talk is to explain the slogan 'trees are to
 polynomial monads as linear orders are to categories', and to make it
 precise.  First I use polynomial functors to give a formal and conceptual
 construction of a category of trees (the Omega of Moerdijk and Weiss), and
 describe its main features.  Then I explain how polynomial endofunctors
 and polynomial monads are obtained by gluing together trees, and give a
 nerve theorem characterising polynomial monads among all presheaves on the
 category of trees.  These constructions and results fit into a general
 machinery developed recently by Weber, but contain some new interesting
 twists due to the fact that the category of polynomial endofunctors is not
 itself a presheaf category.  (Yet polynomial endofunctors have elements
 and canonical diagrams.)  I dedicate this work to my father.